Chapter Nine
Taylor and Laurent Series
9.1. Taylor series. Suppose f is analytic on the open disk |z
z
0
|
r. Let z be any point in
this disk and choose C to be the positively oriented circle of radius
, where
|z
z
0
|
r. Then for sC we have
1
s
z
1
s z
0
z z
0
1
s
z
0
1
1
z
z
0
s
z
0
j
0
z z
0
j
s z
0
j
1
since |
z
z
0
s
z
0
|
1. The convergence is uniform, so we may integrate
C
f
s
s
z ds
j
0
C
f
s
s z
0
j
1
ds
z z
0
j
, or
f
z 1
2
i
C
f
s
s
z ds
j
0
1
2
i
C
f
s
s z
0
j
1
ds
z z
0
j
.
We have thus produced a power series having the given analytic function as a limit:
f
z
j
0
c
j
z z
0
j
, |z
z
0
|
r,
where
c
j
1
2
i
C
f
s
s z
0
j
1
ds.
This is the celebrated Taylor Series for f at z
z
0
.
We know we may differentiate the series to get
f
z
j
1
jc
j
z z
0
j
1
9.1
and this one converges uniformly where the series for f does. We can thus differentiate
again and again to obtain
f
n
z
j
n
j
j 1j 2 j n 1c
j
z z
0
j
n
.
Hence,
f
n
z
0
n!c
n
, or
c
n
f
n
z
0
n!
.
But we also know that
c
n
1
2
i
C
f
s
s z
0
n
1
ds.
This gives us
f
n
z
0
n!
2
i
C
f
s
s z
0
n
1
ds, for n
0, 1, 2, .
This is the famous Generalized Cauchy Integral Formula. Recall that we previously
derived this formula for n
0 and 1.
What does all this tell us about the radius of convergence of a power series? Suppose we
have
f
z
j
0
c
j
z z
0
j
,
and the radius of convergence is R. Then we know, of course, that the limit function f is
analytic for |z
z
0
|
R. We showed that if f is analytic in |z z
0
|
r, then the series
converges for |z
z
0
|
r. Thus r R, and so f cannot be analytic at any point z for which
|z
z
0
|
R. In other words, the circle of convergence is the largest circle centered at z
0
inside of which the limit f is analytic.
9.2
Example
Let f
z expz e
z
. Then f
0 f
0 f
n
0 1, and the Taylor series for f
at z
0
0 is
e
z
j
0
1
j!
z
j
and this is valid for all values of z since f is entire. (We also showed earlier that this
particular series has an infinite radius of convergence.)
Exercises
1. Show that for all z,
e
z
e
j
0
1
j!
z
1
j
.
2. What is the radius of convergence of the Taylor series
j
0
n
c
j
z
j
for tanh z ?
3. Show that
1
1
z
j
0
z i
j
1 i
j
1
for |z
i| 2 .
4. If f
z
1
1
z
, what is f
10
i ?
5. Suppose f is analytic at z
0 and f0 f
0 f
0 0. Prove there is a function g
analytic at 0 such that f
z z
3
g
z in a neighborhood of 0.
6. Find the Taylor series for f
z sin z at z
0
0.
7. Show that the function f defined by
9.3
f
z
sin z
z
for z
0
1
for z
0
is analytic at z
0, and find f
0.
9.2. Laurent series. Suppose f is analytic in the region R
1
|z z
0
|
R
2
, and let C be a
positively oriented simple closed curve around z
0
in this region. (Note: we include the
possiblites that R
1
can be 0, and R
2
.) We shall show that for z C in this region
f
z
j
0
a
j
z z
0
j
j
1
b
j
z z
0
j
,
where
a
j
1
2
i
C
f
s
s z
0
j
1
ds, for j
0, 1, 2,
and
b
j
1
2
i
C
f
s
s z
0
j1
ds, for j
1, 2, .
The sum of the limits of these two series is frequently written
f
z
j
c
j
z z
0
j
,
where
c
j
1
2
i
C
f
s
s z
0
j
1
, j
0, 1, 2, .
This recipe for f
z is called a Laurent series, although it is important to keep in mind that
it is really two series.
9.4
Okay, now let’s derive the above formula. First, let r
1
and r
2
be so that
R
1
r
1
|z z
0
|
r
2
R
2
and so that the point z and the curve C are included in the
region r
1
|z z
0
|
r
2
. Also, let
be a circle centered at z and such that is included in
this region.
Then
f
s
s
z
is an analytic function (of s) on the region bounded by C
1
, C
2
, and
, where C
1
is
the circle |z|
r
1
and C
2
is the circle |z|
r
2
. Thus,
C
2
f
s
s
z ds
C
1
f
s
s
z ds
f
s
s
z ds.
(All three circles are positively oriented, of course.) But
f
s
s
z
ds
2ifz, and so we have
2
ifz
C
2
f
s
s
z ds
C
1
f
s
s
z ds.
Look at the first of the two integrals on the right-hand side of this equation. For s
C
2
, we
have |z
z
0
|
|s z
0
|, and so
1
s
z
1
s z
0
z z
0
1
s
z
0
1
1
z
z
0
s
z
0
1
s
z
0
j
0
z
z
0
s
z
0
j
j
0
1
s z
0
j
1
z z
0
j
.
9.5
Hence,
C
2
f
s
s
z ds
j
0
C
2
f
s
s z
0
j
1
ds
z z
0
j
.
j
0
C
f
s
s z
0
j
1
ds
z z
0
j
For the second of these two integrals, note that for s
C
1
we have |s
z
0
|
|z z
0
|, and so
1
s
z
1
z z
0
s z
0
1
z
z
0
1
1
s
z
0
z
z
0
1
z
z
0
j
0
s
z
0
z
z
0
j
j
0
s z
0
j
1
z z
0
j
1
j
1
s z
0
j
1
1
z z
0
j
j
1
1
s z
0
j1
1
z z
0
j
As before,
C
1
f
s
s
z ds
j
1
C
2
f
s
s z
0
j1
ds
1
z z
0
j
j
1
C
f
s
s z
0
j1
ds
1
z z
0
j
Putting this altogether, we have the Laurent series:
f
z 1
2
i
C
2
f
s
s
z ds
1
2
i
C
1
f
s
s
z ds
j
0
1
2
i
C
f
s
s z
0
j
1
ds
z z
0
j
j
1
1
2
i
C
f
s
s z
0
j1
ds
1
z z
0
j
.
Example
9.6
Let f be defined by
f
z
1
z
z 1
.
First, observe that f is analytic in the region 0
|z| 1. Let’s find the Laurent series for f
valid in this region. First,
f
z
1
z
z 1
1z 1
z
1
.
From our vast knowledge of the Geometric series, we have
f
z 1z
j
0
z
j
.
Now let’s find another Laurent series for f, the one valid for the region 1
|z| .
First,
1
z
1
1
z
1
1
1
z
.
Now since |
1
z
|
1, we have
1
z
1
1
z
1
1
1
z
1z
j
0
z
j
j
1
z
j
,
and so
f
z 1z 1
z
1
1
z
j
1
z
j
f
z
j
2
z
j
.
Exercises
8. Find two Laurent series in powers of z for the function f defined by
9.7
f
z
1
z
2
1 z
and specify the regions in which the series converge to f
z.
9. Find two Laurent series in powers of z for the function f defined by
f
z
1
z
1 z
2
and specify the regions in which the series converge to f
z.
10. Find the Laurent series in powers of z
1 for fz
1
z
in the region 1
|z 1| .
9.8